1999 Fiscal Year Final Research Report Summary
Study of nonlinear prabolic systems and related elliptic systems
| Project/Area Number |
09640228
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | Waseda University |
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Principal Investigator |
YAMADA Yoshio Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (20111825)
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| Co-Investigator(Kenkyū-buntansha) |
TAKEUCHI Shingo JSPS Research Fellow, 特別研究員
NAKASHIMA Kimie Waseda University, School of Sci. & Eng., Assist. Prof., 理工学部, 助手 (10318800)
OTANI Mitsuharu Waseda University, School of Sci. & Eng..,Professor, 理工学部, 教授 (30119656)
TSUTSUMI Masayoshi Waseda University, School of Sci. & Eng., Professor, 理工学部, 教授 (70063774)
HIROSE Munemitsu JSPS Research Fellow, 特別研究員 (50287984)
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| Project Period (FY) |
1997 – 1999
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| Keywords | reaction-diffusion equation / Lotka-Volterra type / positive steady state / bifurcation / stability / p-Laplacian |
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| Research Abstract |
(1) Analysis of reaction diffusion systems with cross-diffusion terms : We have discussed reaction diffusion systems with cross-diffusion and reaction of Lotka-Volterra type. These systems appear in mathematical biology. Mathematically, it is very important to derive sufficient conditions for the existence of time-global solutions and get information on the structure of positive stationary solutions (biologically, coexistence states). As to the non-stationary problem a global existence result has been obtained in one and two space-dimensions. For the stationary problem with zero Dirichlet boundary condition, we have studied uniqueness and non-uniqueness of positive stationary solutions as well as sufficient conditions for their existence. It is proved that our system admits multiple existence of postive solutions. Moreover, numerical simulations exhibit complicate structure of positive stationary solutions such as bifurcation of symmetric solutions from semitrivial solutions and, additionally, bifurcation of asymmetric solutions from symmetric ones.
(2) Analysis of quasilinear parabolic equations with p-Laplacian and logistic terms : Although the nonlinearity and degeneracy of p-Laplacian brings about the difficulty, it also gives remarkable nonlinear phenomenon. We have obtained satisfactory understanding on the structure of stationary solutions in higher space dimension as well as one dimension. In particular, we also have studied profiles of stationary solutions and proved interesting results on flat hats which stem from degenerate diffusion. Furthermore, we could show interesting information on the temporal and spatial change of non-stationary solutions.
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